Lecture 13

04/03/2023, M.

Today

Anderson model
Wave propagation (conduction) as diffusion
Quantum correction

1. Anderson's model (1958)

Anderson's model describes an electron in a 1D lattice with random potential

\begin{matrix} (1) & H = \underset{hopping}{\underset{⏟}{- t \sum_{⟨ x, x^{'} ⟩} c_{x}^{+} c_{x^{'}}}} + \underset{random potential}{\underset{⏟}{\sum_{x} w_{x} c_{x}^{+} c_{x}}} \end{matrix}

$w_x\in [-W,W]$ with the probability distribution

\begin{matrix} (2) & \begin{matrix} P (w) {\begin{cases} \frac{1}{2 W}, & | w | \leq W \\ 0, & otherwise . \end{cases} \end{matrix} \end{matrix}

$W\ll t$ $W\gg t$ $W>0,t=0$ $W/t$ , there must be a transition point beyond which the localized and delocalized states coexist (at different energies).

$W/t$ $t$ $W_c$ $W>W_c$ at some energies, the wavefunction decays strongly (typically exponentially).

$W/t$ $W_c$ , and this is the Anderson transition.

$\xi$ .

In actual calculations, you can use the inverse participation ratio

\begin{matrix} (3) & P_{q} = \int d^{d} r | Ψ (r) |^{2 q} \end{matrix}

$q=1$ $P_1=1$ $q>1$

$|\psi|^2\sim L^{-d}$ $P_q\sim L^{-d(q-1)}$
$|\psi|^2\sim\xi^{-d}$ $\xi^d$ $P_q\sim\xi^{-d(q-1)}$

2. Classical diffusion

Fick's law

\begin{matrix} (4) & j = - D \nabla n \end{matrix}

with equation of continuity

\begin{matrix} (5) & \nabla \cdot j + \partial_{t} n = 0 \end{matrix}

leads to Fick's second law

\begin{matrix} (6) & \partial_{t} n = D \nabla^{2} n \end{matrix}

$r\rightarrow q$

\begin{matrix} (7) & \partial_{t} n (q, t) = - D q^{2} n (q, t) . \end{matrix}

Relating to random walks. The central limit theorem tells us that the sum of a large number of random variables is Gaussian distributed and has a variance (width squared) proportional to the number of steps. This gives the probability density

\begin{matrix} (8) & P (r, t) = (2 π α t)^{- d / 2} e^{- \frac{r^{2}}{2 α t}} \end{matrix}

$\alpha$ is a microscopic parameter (step size and walk speed). Fourier transforming the coordinates, we have

\begin{matrix} (9) & P (q, t) = e^{- \frac{1}{2} α q^{2} t} \end{matrix}

$\alpha=2D$ . Then the probability density

\begin{matrix} (10) & p (r, t) = \frac{n (r, t)}{N} = \frac{e^{- \frac{r^{2}}{4 D t}}}{(4 π D t)^{d / 2}} \end{matrix}

Also introduce the diffusion length

\begin{matrix} (11) & ℓ_{D} = \sqrt{2 d D t} . \end{matrix}

$v_\text F$ . Taking the collision to occur with a Poisson distribution in time

\begin{matrix} (12) & p (τ^{'}) = \frac{1}{τ} e^{- τ^{'} / τ} \end{matrix}

$t/\tau$ , so the mean square displacement is

\begin{matrix} (13) & ⟨ r^{2} ⟩ = Δ^{2} \frac{t}{τ} \end{matrix}

$\Delta$ $r=a_1+a_2+\cdots$ $a_i=\pm\Delta$ .

The mean square step length

\begin{matrix} (14) & Δ^{2} = ⟨ (v_{F} τ^{'})^{2} ⟩ = 2 v_{F}^{2} τ^{2} . \end{matrix}

So we find

\begin{matrix} (15) & ⟨ r^{2} ⟩ = 2 v_{F}^{2} τ t = ℓ_{D}^{2} = 2 d D t \end{matrix}

we have

\begin{matrix} (16) & D = \frac{1}{d} v_{F}^{2} τ = \frac{1}{d} v_{F} ℓ . \end{matrix}

Substitute into the Einstein relation

\begin{matrix} (17) & σ = e^{2} \frac{\partial n}{\partial μ} D = \frac{n e^{2} τ}{m} \end{matrix}

So, we have developed a physical picture that connects the evolution of quantum wave functions with the classical notion of diffusion, which in turn can be viewed as random walks form scatterer to scatter in the sample.

3. Multiple scattering as diffusion

\begin{matrix} (18) & \begin{aligned} ψ (r, t) = ⟨ r, t ∣ 0, 0 ⟩ \\ \int d^{d} r | ψ (r, t) |^{2} = 1 \end{aligned} \end{matrix}

path integral:

\begin{matrix} (19) & \begin{aligned} ψ (r, t) & = \int D (r^{'} (t^{'})] e^{\frac{i}{ℏ} S [r^{'} (t^{'})]} \\ = \frac{1}{A} \sum_{all paths} e^{\frac{i}{ℏ} S [path]} . \\ \approx \frac{1}{A} \sum_{cl. path} e^{\frac{i}{ℏ} S [cl. paths]} \\ = \frac{1}{A} \sum_{j} e^{i θ j} \\ \equiv \sum_{j} A_{j} \end{aligned} \end{matrix}

where the semiclassical is introduced, by summing only over the classical paths. A classical path is one which minimizes the action

\begin{matrix} (20) & {(\frac{δ S}{δ r (t)})}_{r_{cl}} = 0 \end{matrix}

So probability density is

\begin{matrix} (21) & | ψ (r, t) |^{2} = \frac{1}{A^{2}} \sum_{j, k} e^{- i (θ_{k} - θ_{j})} \end{matrix}

Disorder averaging:

\begin{matrix} (22) & \begin{aligned} \overset{―}{| ψ (r, t) |^{2}} & = \frac{1}{A^{2}} \overset{―}{\sum_{R, j} e^{- i (θ_{k} - θ_{j})}} \\ = \frac{1}{A^{2}} N (r, t) \end{aligned} \end{matrix}

$N(r,t)$ $(0,0)$ $(r,t)$ ,

$N(r,t) \sim P(r,t)$ $k^{−d}_F$ , which is needed to convert the Gaussian density into a dimensionless number, can be viewed as the fuzzy quantum “size” of the particle

\begin{matrix} (23) & \overset{―}{| Ψ |^{2}} \approx \frac{N (\vec{r}, t)}{A^{2}} = \frac{k_{F}^{- d}}{A^{2}} P (r, t) = (4 π D t)^{- d / 2} e^{- \frac{r^{2}}{4 D t}}, \end{matrix}

So a semiclassical propagation of quantum waves in a random medium can be described as diffusion.

4. Quantum correction

$j$ $\tilde j$

\begin{matrix} (24) & \begin{matrix} j : r (t^{'}), t^{'} \in [0, t] \\ \tilde{j} : r (t - t^{'}), t^{'} \in [0, t] \end{matrix} \end{matrix}

Time-reversal symmetry means the action does not change under time reversal

\begin{matrix} (25) & S [r (t^{'})] = S [r (t - t^{'})] \end{matrix}

\begin{matrix} (26) & A_{j} = A_{\tilde{j}} \end{matrix}

The probability of return is

\begin{matrix} \begin{matrix} \begin{aligned} | ψ (0, t) |^{2} & = {| \sum_{j = 1}^{N / 2} (A_{j} + A_{\tilde{j}}) |}^{2} \\ = 4 {| \sum_{j = 1}^{N / 2} A_{j} |}^{2} \\ = 4 (\sum_{j}^{N / 2} | A_{j} |^{2} + \sum_{j \neq k}^{N / 2} A_{j}^{*} A_{k}) \end{aligned} \end{matrix} \end{matrix}

$A_j$ $A_k$ $j\neq k$ have no definite phase relation, the sum should vanish upon disorder averaging. We find the quantum probability of return to be

\begin{matrix} (27) & | ψ (0, t) |^{2} = 2 \sum_{j = 1}^{N} | A_{j} |^{2} = 2 \times P (0, t) \end{matrix}

This is the picture of electron localization via quantum interference: increased probability of return.

$k$ $k'$ :

\begin{matrix} (28) & f (k^{'}, k) = - \frac{m}{2 π ℏ} \sum_{n = 0}^{\infty} \sum_{k_{1} \dots k_{n}} m_{k^{'} k} (k_{n} \dots k_{1}) \end{matrix}

where

\begin{matrix} (29) & M_{k^{'} k^{'}} (k_{n} \dots k_{1}) = v (k^{'} - k_{n}) G (k_{n}) v (k_{n} - k_{n - 1}) \dots G (k_{1}) v (k_{1} - k) \end{matrix}

$k'=-k$

\begin{matrix} (30) & M_{- k, k} (k_{n} \dots k_{1}) = v (- k - k_{n}) G (k_{n}) \dots G (k_{1}) v (k_{1} - k) \end{matrix}

$\left(k_n, k_{n-1} \cdots k_1\right) \rightarrow\left(-k_1,-k_2 \cdots,-k_n\right)$

\begin{matrix} (31) & \begin{aligned} M_{- k, k} (- k_{1}, - k_{2}, \dots - k_{n}) \\ = v (- k + k_{1}) G (- k_{1}) v (- k_{1} + k_{2}) \dots G (- k_{n}) v (- k_{n} - k) \end{aligned} \end{matrix}

$G(\Theta k)=G(-k)=G(k)$ $p_n$

\begin{matrix} (32) & M_{- k, k} (p_{n}) = M_{- k, k} (Θ p_{n}) \end{matrix}

So in the k-space, the amplitudes of backscattering via time-reversed paths are the same. Therefore, they will survive disorder averaging.

To sum up, the quantum correction to semiclassical diffusion

real space: increased probability of return
k-space: enhanced back scattering